Characteristic classes and transversality

Marcelo A. Aguilar, José Luis Cisneros-Molina*, Martín Eduardo Frías-Armenta

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let ξ be a smooth vector bundle over a differentiable manifold M. Let h : εn - i + 1 → ξ be a generic bundle morphism from the trivial bundle of rank n - i + 1 to ξ. We give a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold over(Z, ̃) (h) and a map φ{symbol} : over(Z, ̃) (h) → M whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by φ{symbol}, of the fundamental class of the manifold over(Z, ̃) (h). We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.

Original languageEnglish
Pages (from-to)1220-1235
Number of pages16
JournalTopology and its Applications
Volume154
Issue number7 SPEC. ISS.
DOIs
StatePublished - 1 Apr 2007

Bibliographical note

Funding Information:
* Corresponding author. E-mail addresses: marcelo@matem.unam.mx (M.A. Aguilar), jlcm@matcuer.unam.mx (J.L. Cisneros-Molina), eduardo@gauss.mat.uson.mx (M.E. Frías-Armenta). 1 Partially supported by Proyecto CONACyT G36357-E.

Keywords

  • Characteristic classes
  • Generic bundle morphisms

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